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Theorem sepnsepolem2 48049
Description: Open neighborhood and neighborhood is equivalent regarding disjointness. Lemma for sepnsepo 48050. Proof could be shortened by 1 step using ssdisjdr 47987. (Contributed by Zhi Wang, 1-Sep-2024.)
Hypothesis
Ref Expression
sepnsepolem2.1 (πœ‘ β†’ 𝐽 ∈ Top)
Assertion
Ref Expression
sepnsepolem2 (πœ‘ β†’ (βˆƒπ‘¦ ∈ ((neiβ€˜π½)β€˜π·)(π‘₯ ∩ 𝑦) = βˆ… ↔ βˆƒπ‘¦ ∈ 𝐽 (𝐷 βŠ† 𝑦 ∧ (π‘₯ ∩ 𝑦) = βˆ…)))
Distinct variable groups:   𝑦,𝐷   𝑦,𝐽   π‘₯,𝑦
Allowed substitution hints:   πœ‘(π‘₯,𝑦)   𝐷(π‘₯)   𝐽(π‘₯)

Proof of Theorem sepnsepolem2
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 sepnsepolem2.1 . 2 (πœ‘ β†’ 𝐽 ∈ Top)
2 id 22 . . 3 (𝐽 ∈ Top β†’ 𝐽 ∈ Top)
3 sslin 4230 . . . . 5 (𝑧 βŠ† 𝑦 β†’ (π‘₯ ∩ 𝑧) βŠ† (π‘₯ ∩ 𝑦))
4 sseq0 4396 . . . . . 6 (((π‘₯ ∩ 𝑧) βŠ† (π‘₯ ∩ 𝑦) ∧ (π‘₯ ∩ 𝑦) = βˆ…) β†’ (π‘₯ ∩ 𝑧) = βˆ…)
54ex 411 . . . . 5 ((π‘₯ ∩ 𝑧) βŠ† (π‘₯ ∩ 𝑦) β†’ ((π‘₯ ∩ 𝑦) = βˆ… β†’ (π‘₯ ∩ 𝑧) = βˆ…))
63, 5syl 17 . . . 4 (𝑧 βŠ† 𝑦 β†’ ((π‘₯ ∩ 𝑦) = βˆ… β†’ (π‘₯ ∩ 𝑧) = βˆ…))
76adantl 480 . . 3 ((𝐽 ∈ Top ∧ 𝑧 βŠ† 𝑦) β†’ ((π‘₯ ∩ 𝑦) = βˆ… β†’ (π‘₯ ∩ 𝑧) = βˆ…))
8 ineq2 4201 . . . . 5 (𝑦 = 𝑧 β†’ (π‘₯ ∩ 𝑦) = (π‘₯ ∩ 𝑧))
98eqeq1d 2727 . . . 4 (𝑦 = 𝑧 β†’ ((π‘₯ ∩ 𝑦) = βˆ… ↔ (π‘₯ ∩ 𝑧) = βˆ…))
109adantl 480 . . 3 ((𝐽 ∈ Top ∧ 𝑦 = 𝑧) β†’ ((π‘₯ ∩ 𝑦) = βˆ… ↔ (π‘₯ ∩ 𝑧) = βˆ…))
112, 7, 10opnneieqv 48037 . 2 (𝐽 ∈ Top β†’ (βˆƒπ‘¦ ∈ ((neiβ€˜π½)β€˜π·)(π‘₯ ∩ 𝑦) = βˆ… ↔ βˆƒπ‘¦ ∈ 𝐽 (𝐷 βŠ† 𝑦 ∧ (π‘₯ ∩ 𝑦) = βˆ…)))
121, 11syl 17 1 (πœ‘ β†’ (βˆƒπ‘¦ ∈ ((neiβ€˜π½)β€˜π·)(π‘₯ ∩ 𝑦) = βˆ… ↔ βˆƒπ‘¦ ∈ 𝐽 (𝐷 βŠ† 𝑦 ∧ (π‘₯ ∩ 𝑦) = βˆ…)))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 394   = wceq 1533   ∈ wcel 2098  βˆƒwrex 3060   ∩ cin 3940   βŠ† wss 3941  βˆ…c0 4319  β€˜cfv 6543  Topctop 22808  neicnei 23014
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5281  ax-sep 5295  ax-nul 5302  ax-pow 5360  ax-pr 5424
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-ral 3052  df-rex 3061  df-reu 3365  df-rab 3420  df-v 3465  df-sbc 3771  df-csb 3887  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4320  df-if 4526  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4905  df-iun 4994  df-br 5145  df-opab 5207  df-mpt 5228  df-id 5571  df-xp 5679  df-rel 5680  df-cnv 5681  df-co 5682  df-dm 5683  df-rn 5684  df-res 5685  df-ima 5686  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-top 22809  df-nei 23015
This theorem is referenced by:  sepnsepo  48050
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